Problem 65 - University Calculus: Early Transcendentals

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\begin{aligned} &\int\left(1+\tan ^{2} \theta\right) d \theta\\\ &\text { (Hint: }\left.1+\tan ^{2} \theta=\sec ^{2} \theta\right) \end{aligned}$$

Step-by-Step Solution

Answer

The antiderivative is \( \tan \theta + C \).

Step 1: Recognize the Hint

We are given the integral \( \int (1 + \tan^2 \theta) \, d\theta \). Based on the hint, rewrite the expression as \( \int \sec^2 \theta \, d\theta \). This change uses the trigonometric identity \( 1 + \tan^2 \theta = \sec^2 \theta \).

Step 2: Find the Indefinite Integral

The integral \( \int \sec^2 \theta \, d\theta \) is a standard integral that equals \( \tan \theta + C \), where \( C \) is the constant of integration.

Step 3: Verify the Solution by Differentiation

Differentiate the solution \( \tan \theta + C \) with respect to \( \theta \). The derivative of \( \tan \theta \) is \( \sec^2 \theta \), and the derivative of the constant \( C \) is 0. Hence, \( \frac{d}{d\theta}(\tan \theta + C) = \sec^2 \theta \), which matches the integrand.

Key Concepts

Trigonometric Identities

In calculus, trigonometric identities are pivotal, especially when solving integrals. Recognizing specific identities can simplify complex expressions, making integration much easier. In the given exercise, the identity \( 1 + \tan^2 \theta = \sec^2 \theta \) plays a crucial role. This identity allows us to convert the integrand from a form involving \( \tan \theta \) to one with \( \sec^2 \theta \), which is much more straightforward to integrate.

Understanding these basic identities not only helps in reducing the complexity of the integral but also connects different trigonometric functions through their relationships. Here are some critical trigonometric identities to have at your fingertips:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• \( 1 + \tan^2 \theta = \sec^2 \theta \)
• \( 1 + \cot^2 \theta = \csc^2 \theta \)

These identities are used extensively in calculus to simplify problems and should be committed to memory or readily referenced during integration tasks.

Antiderivatives

Antiderivatives, often synonymous with indefinite integrals, involve finding a function whose derivative equals the given function. In essence, it answers the question: If one function is a derivative, what was the original function?

In our exercise, after using the trigonometric identity, we arrive at the integral \( \int \sec^2 \theta \, d\theta \). The antiderivative of \( \sec^2 \theta \) is known to be \( \tan \theta \), because the derivative of \( \tan \theta \) is \( \sec^2 \theta \). This pattern is essential in solving integrals, as many popular functions have well-known antiderivatives that can be directly applied when encountered.

Remember, every indefinite integral will have a constant of integration, \( C \), because differentiation of a constant is zero. As a result, the general antiderivative of \( \sec^2 \theta \) becomes \( \tan \theta + C \).

• Standard Antiderivative Example: \( \int e^x \, dx = e^x + C \)
• Rationale: Because derivative of \( e^x \) with respect to \( x \) is \( e^x \).

Consistently practicing these patterns will sharpen your intuitiveness in identifying antiderivatives.

Integration Verification

Integration verification is an invaluable step in calculus to ensure that your solution, i.e., the antiderivative, is correct. This is particularly important when you're crafting a solution independently or want to double-check your work.

After finding the antiderivative, the next step is to differentiate it back and check if the original function is retrieved. This backward-step process solidifies understanding and confirms that computations are error-free. Using the exercise's results, differentiating our answer \( \tan \theta + C \) gives \( \sec^2 \theta \), which matches the original expression we integrated, \( 1 + \tan^2 \theta \) simplified to \( \sec^2 \theta \).

• Verification Steps: Differentiate the antiderivative.
• Check if the result matches the initial integrand.

Practicing integration followed by verification not only builds confidence but also ensures thoroughness, especially in more complex integrals where errors might easily creep in.